Symmetric Difference

Calculator for computing symmetric differences with comprehensive formulas and examples

Symmetric Difference Calculator

What is calculated?

The symmetric difference A ⊕ B (also A △ B) contains all elements that are either in A or in B, but not in both. It corresponds to the logical XOR (exclusive or).

Enter the Sets

Values separated by spaces or semicolons

Values separated by spaces or semicolons
Result (A ⊕ B)
Symmetric Difference:
Elements that are in only one of the two sets

Symmetric Difference Info

Properties

Symmetric Difference A ⊕ B:

  • Commutative: A ⊕ B = B ⊕ A
  • Associative: (A ⊕ B) ⊕ C = A ⊕ (B ⊕ C)
  • A ⊕ A = ∅ (Self-cancellation)
  • A ⊕ ∅ = A (Neutral element)
Symmetric Difference Venn Diagram

Venn diagram of symmetric difference

Memory aid: "Either A or B, but not both simultaneously."

XOR Logic
A B A ⊕ B
000
101
011
110
Examples
Standard:
{1,2,3} ⊕ {2,3,4} = {1,4}
Disjoint:
{1,2} ⊕ {3,4} = {1,2,3,4}
Identical:
{1,2,3} ⊕ {1,2,3} = ∅
Related Operations

→ Set Union
→ Set Intersection
→ Set Difference


Symmetric Difference Formulas

Basic Definition
\[A \oplus B = (A \setminus B) \cup (B \setminus A)\] Union of differences
Alternative Definition
\[A \triangle B = (A \cup B) \setminus (A \cap B)\] Union minus intersection
Set Notation
\[A \oplus B = \{x : (x \in A \land x \notin B) \lor (x \in B \land x \notin A)\}\] Logical definition
XOR Representation
\[A \oplus B = \{x : x \in A \oplus x \in B\}\] Exclusive or
Commutative Law
\[A \oplus B = B \oplus A\] Order doesn't matter
Associative Law
\[(A \oplus B) \oplus C = A \oplus (B \oplus C)\] Grouping doesn't matter

Detailed Calculation Example

Example: A = {1,2,3,4,5}, B = {4,5,6,7,8,9}

Given:

  • A = {1, 2, 3, 4, 5}
  • B = {4, 5, 6, 7, 8, 9}

Step 1 - Individual differences:

A \ B = {1, 2, 3}
B \ A = {6, 7, 8, 9}

Step 2 - Union:

A ⊕ B = {1,2,3,6,7,8,9}

Alternative calculation:

A ∪ B = {1,2,3,4,5,6,7,8,9}
A ∩ B = {4,5}
(A ∪ B) \ (A ∩ B) = {1,2,3,6,7,8,9} ✓

Interpretation: The symmetric difference contains all elements that are either only in A or only in B, but not in both.

XOR Logic and Applications

Symmetric Difference as XOR Operation

Digital Logic:

In Boolean algebra, the symmetric difference corresponds to the XOR operation (exclusive or). The result is only "true" when exactly one of the two inputs is "true".

Practical Example:

Lights in a room:
Switch A: {On}
Switch B: {On}
XOR circuit: Light is OFF (both switches on = off)

Cryptography Application

Simple XOR encryption:

Plaintext:
{1,0,1,1,0}
Key:
{1,1,0,1,1}
Ciphertext:
{0,1,1,0,1}

Property: Ciphertext ⊕ Key = Plaintext (Self-inverse)

Algebraic Properties

Fundamental Properties of Symmetric Difference
Self-Cancellation
\[A \oplus A = \emptyset\]

Set with itself yields empty set

Neutral Element
\[A \oplus \emptyset = A\]

Empty set is neutral element

Self-Inverse
\[A \oplus B \oplus B = A\]

Every set is its own inverse

Distributivity
\[A \cap (B \oplus C) = (A \cap B) \oplus (A \cap C)\]

Distribution over intersection

Practical Example for Self-Inverse

Encryption and Decryption:

Plaintext: A = {1,3,5}
Key: B = {2,3,4}
Encrypted: A ⊕ B = {1,2,4,5}
Decrypted: (A ⊕ B) ⊕ B = {1,3,5} = A ✓

Mathematical Properties

Basic Properties
  • Commutativity: A ⊕ B = B ⊕ A
  • Associativity: (A ⊕ B) ⊕ C = A ⊕ (B ⊕ C)
  • Self-inverse: A ⊕ A = ∅
  • Neutral element: A ⊕ ∅ = A
Special Properties
  • Group structure: (P(U), ⊕) is abelian group
  • Cardinality: |A ⊕ B| = |A| + |B| - 2|A ∩ B|
  • Disjoint sets: A ⊕ B = A ∪ B when A ∩ B = ∅
  • Identical sets: A ⊕ A = ∅
Important Notes

XOR property: Symmetric difference corresponds to exclusive or

Cryptography: Foundation for many encryption methods

Practical Applications

Cryptography
  • XOR encryption
  • One-time pad
  • Stream ciphers
  • Key exchange protocols
Digital Technology
  • XOR gates
  • Parity checking
  • Adder circuits
  • Error correction
Data Analysis
  • Finding differences
  • Exclusive features
  • Change detection
  • Difference analysis




Set theory functions

Complement  •  Difference  •  Intersection  •  Symmetric Difference  •  Union  •